Question

What is the square root of \[\dfrac{{64}}{{125}}\]?

Answer 1

Hint: Square root of a number is a value, which on multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as \[x = \sqrt y \] or we can express the same equation as \[{x^2} = y\]. Here we need to find the square root of the numerator and denominator as well. We solve this using the factorization method.

Complete step-by-step solutions:
Given,
\[\sqrt {\dfrac{{64}}{{125}}} \]
Now 64 can be factorized as,
\[64 = 1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
We can see that 2 is multiplied twice three times, we multiply that we get,
\[64 = {2^2} \times {2^2} \times {2^2}\].
Now 125 can be factored as
 \[125 = 1 \times 5 \times 5 \times 5\]
we can see that 5 is multiplied twice we have,
\[125 = 1 \times 5 \times {5^2}\]
Then,
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = \sqrt {\dfrac{{{2^2} \times {2^2} \times {2^2}}}{{5 \times {5^2}}}} \]
Now we know that square and square root will get cancel and we take term out the radical symbol,
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = \dfrac{{2 \times 2 \times 2}}{{5\sqrt 5 }}\].
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = \dfrac{8}{{5\sqrt 5 }}\]. This is the required result.
We can put it in decimal form if we want,
We know \[\sqrt 5 = 2.23\] then,
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = \dfrac{8}{{5 \times 2.23}}\]
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = \dfrac{8}{{11.15}}\]
\[ \Rightarrow \sqrt {\dfrac{{64}}{{125}}} = 0.7174\]. This is the decimal form.

Thus the final answer is = 0.7174.

Note: Here \[\sqrt {} \] is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.